A topological Maslov index for 3-graded Lie groups - ResearchGate

A topological Maslov index for 3-graded Lie groups 45
This implies that [h, [g1, g−1]] = {0} and in particular [h, [x, g−1]] = {0}.
Since the map (adx) 2 : g−1 → g1 is bijective,
ad x| [x,g−1]: [x, g−1] → g1
also is bijective. Thus adx([x, g−1]) = g1 and hence
g0 = [x, g−1] ⊕ (keradx ∩ g0).
For z ∈ g0 ∩ ker adx the operators adz and adx commutes, so that
(adx) 2 ([y, z]) ∈ −adz(ad x) 2 .y = −2 ad z.x = 0,
and therefore [y, z] = 0. This also implies that [h, z] = 0, and we conclude that
h ∈ z(g0). Hence 1
2h is a grading element.
(2) For w ∈ g−1 the relations [y, w] = 0 and [h, w] = −2w imply that
w generates an at most 3-dimensional submodule for the Lie subalgebra gx :=
spanK{x, y, h} ([Bou90, Ch. VIII, §1, no. 1, Lemma 1]).
For n = 2 we get with (C.3) and [y, w] = 0:
ad y(adx) 2 .w = [ady, (adx) 2 ].w = −2(ad x) adh + id .w = 2 adx.w.
w. We
conclude that (adx) 2 |g−1 is injective.
For w ∈ g1 the relations [h, w] = 2w and [x, w] = 0, together with the
relation
(C.4)
[adx, (ady) n ] = n(ad y) n−1 ad h − (n − 1) id = n ad h + (n − 1) id (ady) n−1
Therefore (adx) 2 .w = 0 implies [x, w] = 0, so that 0 = [h, w] = −1 2
leads to
(ad x) 2 (ady) 2 .w = (adx).[adx, (ady) 2 ].w = (adx).(2 ady.w) = 2 ad[x, y].w = 4w,
and hence to w ∈ (adx) 2 (g−1).
References
[ASS71] Araki, H., M.-S. Bae Smith, and L. Smith, On the homotopical
significance of the type of von Neumann algebra factors, Commun.
math. Phys. 22 (1971), 71–88.
[BN04a] Bertram, W., and K.-H. Neeb, Projective completions of Jordan
pairs. Part I. Geometries associated to 3-graded Lie algebras, Journal
of Algebra 277:2 (2004), 474–519.
[BN04b] —, Projective completions of Jordan pairs. Part II. Manifold structures
and symmetric spaces, Geom. Dedicata, to appear.